Suppose that X1,X2,...,Xn form a random sample from a uniform distribution on the interval [θ1,θ2], where...

Suppose that X1,X2,...,Xn form a random sample from a uniform distribution on the interval [θ1,θ2], where (−∞ < θ1 < θ2 < ∞). Find MME for both θ1 and θ2.

Expert Solution

orchestra answered 1 year ago

Let X1,X2,...Xn be a random sample of size n form a uniform distribution on the interval...

Let X1,X2,...Xn be a random sample of size n form a uniform distribution on the interval [θ1,θ2]. Let Y = min (X1,X2,...,Xn). (a) Find the density function for Y. (Hint: find the cdf and then differentiate.) (b) Compute the expectation of Y. (c) Suppose θ1= 0. Use part (b) to give an unbiased estimator for θ2.

Suppose that X1, ..., Xn form a random sample from a uniform distribution for on the...

Suppose that X1, ..., Xn form a random sample from a uniform distribution for on the interval [0, θ]. Show that T = max(X1, ..., Xn) is a sufficient statistic for θ.

Let X1,X2,...,Xn be a random sample from a uniform distribution on the interval (0,a). Recall that...

Let X1,X2,...,Xn be a random sample from a uniform distribution on the interval (0,a). Recall that the maximum likelihood estimator (MLE) of a is ˆ a = max(Xi). a) Let Y = max(Xi). Use the fact that Y ≤ y if and only if each Xi ≤ y to derive the cumulative distribution function of Y. b) Find the probability density function of Y from cdf. c) Use the obtained pdf to show that MLE for a (ˆ a =...

Let X1,X2, . . . , Xn be a random sample from the uniform distribution with...

Let X1,X2, . . . , Xn be a random sample from the uniform distribution with pdf f(x; θ1, θ2) = 1/(2θ2), θ1 − θ2 < x < θ1 + θ2, where −∞ < θ1 < ∞ and θ2 > 0, and the pdf is equal to zero elsewhere. (a) Show that Y1 = min(Xi) and Yn = max(Xi), the joint sufficient statistics for θ1 and θ2, are complete. (b) Find the MVUEs of θ1 and θ2.

2. Let X1, ..., Xn be a random sample from a uniform distribution on the interval...

2. Let X1, ..., Xn be a random sample from a uniform distribution on the interval (0, θ) where θ > 0 is a parameter. The prior distribution of the parameter has the pdf f(t) = βαβ/t^(β−1) for α < t < ∞ and zero elsewhere, where α > 0, β > 0. Find the Bayes estimator for θ. Describe the usefulness and the importance of Bayesian estimation. We are assuming that theta = t, but we are unsure if...

Let X1, . . . , Xn be a random sample from a uniform distribution on...

Let X1, . . . , Xn be a random sample from a uniform distribution on the interval [a, b] (i) Find the moments estimators of a and b. (ii) Find the MLEs of a and b.

Suppose X1, X2, ..., Xn is a random sample from a Poisson distribution with unknown parameter...

Suppose X1, X2, ..., Xn is a random sample from a Poisson distribution with unknown parameter µ. a. What is the mean and variance of this distribution? b. Is X1 + 2X6 − X8 an estimator of µ? Is it a good estimator? Why or why not? c. Find the moment estimator and MLE of µ. d. Show the estimators in (c) are unbiased. e. Find the MSE of the estimators in (c). Given the frequency table below: X 0...

Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution...

Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution with pdf f(x; θ) = (x/θ) * e^x^2/(2θ) for x > 0 (a) It can be shown that E(X2 P ) = 2θ. Use this fact to construct an unbiased estimator of θ based on n i=1 X2 i . (b) Estimate θ from the following n = 10 observations on vibratory stress of a turbine blade under specified conditions: 16.88 10.23 4.59 6.66...

Let X1, X2, ..., Xn be a random sample of size from a distribution with probability...

Let X1, X2, ..., Xn be a random sample of size from a distribution with probability density function f(x) = λxλ−1 , 0 < x < 1, λ > 0 a) Get the method of moments estimator of λ. Calculate the estimate when x1 = 0.1, x2 = 0.2, x3 = 0.3. b) Get the maximum likelihood estimator of λ. Calculate the estimate when x1 = 0.1, x2 = 0.2, x3 = 0.3.

. Let X1, X2, ..., Xn be a random sample of size 75 from a distribution...

. Let X1, X2, ..., Xn be a random sample of size 75 from a distribution whose probability distribution function is given by f(x; θ) = ( 1 for 0 < x < 1, 0 otherwise. Use the central limit theorem to approximate P(0.45 < X < 0.55)

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